Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — complement, union and intersection
U={1,2,3,…,10}, A is the set of even numbers in U, and B is the set of multiples of 3 in U. List A∩B, A∪B and A′.
1A={2,4,6,8,10} and B={3,6,9}List both sets out before doing anything with them.
2A∩B={6}Only 6 appears in both — it is the one number under 10 that is both even and a multiple of 3.
3A∪B={2,3,4,6,8,9,10}Everything in either set, with 6 written once only.
4A′={1,3,5,7,9}Everything in U that is not in A — the odd numbers.
Example 2 — a two-set counting problem
In a class of 40 students, 25 study French, 18 study Spanish and 8 study both. How many study at least one of the languages, and how many study neither?
1n(F∪S)=n(F)+n(S)−n(F∩S)Adding alone would count the 8 twice.
2=25+18−8=3535 study at least one language.
3Neither =40−35=5The whole class, minus those doing at least one.
4Check the four regions: French only 17, both 8, Spanish only 10, neither 5.17+8+10+5=40 ✓ — a Venn diagram must account for everybody exactly once.
Example 3 — running the counting formula backwards
Of 50 people surveyed, 30 like tea, 25 like coffee and 5 like neither. How many like both?
1Those liking at least one =50−5=45Start by removing the ones outside both circles.
2n(T∪C)=n(T)+n(C)−n(T∩C)The same formula, with the unknown now inside it.
345=30+25−xLet x be the number who like both.
445=55−x, so x=10Ten people like both.
5Check: tea only 20, both 10, coffee only 15, neither 5 — total 50 ✓Always add the regions back up.